3.2.45 \(\int \frac {-2-x+2 x^4}{(1+x+x^4) \sqrt {1+x+x^2+x^4}} \, dx\) [145]
Optimal. Leaf size=18 \[ -2 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x+x^2+x^4}}\right ) \]
[Out]
-2*arctanh(x/(x^4+x^2+x+1)^(1/2))
________________________________________________________________________________________
Rubi [F]
time = 0.33, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-2-x+2 x^4}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(-2 - x + 2*x^4)/((1 + x + x^4)*Sqrt[1 + x + x^2 + x^4]),x]
[Out]
2*Defer[Int][1/Sqrt[1 + x + x^2 + x^4], x] - 4*Defer[Int][1/((1 + x + x^4)*Sqrt[1 + x + x^2 + x^4]), x] - 3*De
fer[Int][x/((1 + x + x^4)*Sqrt[1 + x + x^2 + x^4]), x]
Rubi steps
\begin {align*} \int \frac {-2-x+2 x^4}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx &=\int \left (\frac {2}{\sqrt {1+x+x^2+x^4}}-\frac {4+3 x}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {1+x+x^2+x^4}} \, dx-\int \frac {4+3 x}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx\\ &=2 \int \frac {1}{\sqrt {1+x+x^2+x^4}} \, dx-\int \left (\frac {4}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}}+\frac {3 x}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {1+x+x^2+x^4}} \, dx-3 \int \frac {x}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx-4 \int \frac {1}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 18, normalized size = 1.00 \begin {gather*} -2 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x+x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-2 - x + 2*x^4)/((1 + x + x^4)*Sqrt[1 + x + x^2 + x^4]),x]
[Out]
-2*ArcTanh[x/Sqrt[1 + x + x^2 + x^4]]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 7.31, size = 4880, normalized size = 271.11
| | |
| method |
result |
size |
| | |
| trager |
\(\ln \left (-\frac {-x^{4}+2 \sqrt {x^{4}+x^{2}+x +1}\, x -2 x^{2}-x -1}{x^{4}+x +1}\right )\) |
\(41\) |
| default |
\(\text {Expression too large to display}\) |
\(4880\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(4880\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x^4-x-2)/(x^4+x+1)/(x^4+x^2+x+1)^(1/2),x,method=_RETURNVERBOSE)
[Out]
4*(RootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^4+_Z^2+_Z+1,index=4))*((RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4
+_Z^2+_Z+1,index=2))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=1))/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+_Z+1,
index=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2)*(x-RootOf(_Z^4+_Z^2+_Z+1,index=2))^2*((RootOf(_Z^4+_Z^2+_Z
+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=1))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=3))/(RootOf(_Z^4+_Z^2+_Z+1,index=3)
-RootOf(_Z^4+_Z^2+_Z+1,index=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2)*((RootOf(_Z^4+_Z^2+_Z+1,index=2)-Ro
otOf(_Z^4+_Z^2+_Z+1,index=1))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=4))/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_
Z^2+_Z+1,index=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2)/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+
_Z+1,index=2))/(RootOf(_Z^4+_Z^2+_Z+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=1))/((x-RootOf(_Z^4+_Z^2+_Z+1,index
=1))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=2))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=3))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=4)))
^(1/2)*EllipticF(((RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+_Z+1,index=2))*(x-RootOf(_Z^4+_Z^2+_Z+1,ind
ex=1))/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+_Z+1,index=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/
2),((RootOf(_Z^4+_Z^2+_Z+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=3))*(RootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^
4+_Z^2+_Z+1,index=4))/(RootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^4+_Z^2+_Z+1,index=3))/(RootOf(_Z^4+_Z^2+_Z+1,i
ndex=2)-RootOf(_Z^4+_Z^2+_Z+1,index=4)))^(1/2))+2*sum(_alpha*(RootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^4+_Z^2+
_Z+1,index=4))*((RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+_Z+1,index=2))*(x-RootOf(_Z^4+_Z^2+_Z+1,index
=1))/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+_Z+1,index=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2)
*(x-RootOf(_Z^4+_Z^2+_Z+1,index=2))^2*((RootOf(_Z^4+_Z^2+_Z+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=1))*(x-Root
Of(_Z^4+_Z^2+_Z+1,index=3))/(RootOf(_Z^4+_Z^2+_Z+1,index=3)-RootOf(_Z^4+_Z^2+_Z+1,index=1))/(x-RootOf(_Z^4+_Z^
2+_Z+1,index=2)))^(1/2)*((RootOf(_Z^4+_Z^2+_Z+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=1))*(x-RootOf(_Z^4+_Z^2+_
Z+1,index=4))/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+_Z+1,index=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2
)))^(1/2)/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+_Z+1,index=2))/(RootOf(_Z^4+_Z^2+_Z+1,index=2)-Root
Of(_Z^4+_Z^2+_Z+1,index=1))/((x-RootOf(_Z^4+_Z^2+_Z+1,index=1))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=2))*(x-RootOf(_
Z^4+_Z^2+_Z+1,index=3))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=4)))^(1/2)*(-RootOf(_Z^4+_Z^2+_Z+1,index=2)^3*_alpha^3+
RootOf(_Z^4+_Z^2+_Z+1,index=2)^3*_alpha^2+RootOf(_Z^4+_Z^2+_Z+1,index=2)^2*_alpha^3-RootOf(_Z^4+_Z^2+_Z+1,inde
x=2)*_alpha^3-RootOf(_Z^4+_Z^2+_Z+1,index=2)^3+RootOf(_Z^4+_Z^2+_Z+1,index=2)*_alpha^2+RootOf(_Z^4+_Z^2+_Z+1,i
ndex=2)^2+_alpha^2-2*RootOf(_Z^4+_Z^2+_Z+1,index=2)-_alpha)*(EllipticF(((RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf
(_Z^4+_Z^2+_Z+1,index=2))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=1))/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+
_Z+1,index=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2),((RootOf(_Z^4+_Z^2+_Z+1,index=2)-RootOf(_Z^4+_Z^2+_Z+
1,index=3))*(RootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^4+_Z^2+_Z+1,index=4))/(RootOf(_Z^4+_Z^2+_Z+1,index=1)-Ro
otOf(_Z^4+_Z^2+_Z+1,index=3))/(RootOf(_Z^4+_Z^2+_Z+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=4)))^(1/2))+(RootOf(
_Z^4+_Z^2+_Z+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=1))*(-RootOf(_Z^4+_Z^2+_Z+1,index=1)^3*_alpha^3+RootOf(_Z^
4+_Z^2+_Z+1,index=1)^3*_alpha^2+RootOf(_Z^4+_Z^2+_Z+1,index=1)^2*_alpha^3-RootOf(_Z^4+_Z^2+_Z+1,index=1)*_alph
a^3-RootOf(_Z^4+_Z^2+_Z+1,index=1)^3+RootOf(_Z^4+_Z^2+_Z+1,index=1)*_alpha^2+RootOf(_Z^4+_Z^2+_Z+1,index=1)^2+
_alpha^2-2*RootOf(_Z^4+_Z^2+_Z+1,index=1)-_alpha)*EllipticPi(((RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2
+_Z+1,index=2))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=1))/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+_Z+1,index
=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2),186/257*RootOf(_Z^4+_Z^2+_Z+1,index=1)^3+284/257*RootOf(_Z^4+_Z
^2+_Z+1,index=1)-191/257*RootOf(_Z^4+_Z^2+_Z+1,index=4)+175/257*RootOf(_Z^4+_Z^2+_Z+1,index=2)-132/257*_alpha^
3*RootOf(_Z^4+_Z^2+_Z+1,index=1)*RootOf(_Z^4+_Z^2+_Z+1,index=2)+78/257*_alpha^2*RootOf(_Z^4+_Z^2+_Z+1,index=4)
*RootOf(_Z^4+_Z^2+_Z+1,index=2)^2-55/257*_alpha^2*RootOf(_Z^4+_Z^2+_Z+1,index=4)*RootOf(_Z^4+_Z^2+_Z+1,index=2
)-239/257*_alpha^2*RootOf(_Z^4+_Z^2+_Z+1,index=4)*RootOf(_Z^4+_Z^2+_Z+1,index=1)^3+6/257*_alpha^2*RootOf(_Z^4+
_Z^2+_Z+1,index=4)*RootOf(_Z^4+_Z^2+_Z+1,index=1)^2-191/257*_alpha^2*RootOf(_Z^4+_Z^2+_Z+1,index=4)*RootOf(_Z^
4+_Z^2+_Z+1,index=1)-154/257*_alpha^2*RootOf(_Z^4+_Z^2+_Z+1,index=2)^2*RootOf(_Z^4+_Z^2+_Z+1,index=1)^3+120/25
7*_alpha^2*RootOf(_Z^4+_Z^2+_Z+1,index=2)^2*RootOf(_Z^4+_Z^2+_Z+1,index=1)^2-222/257*_alpha^2*RootOf(_Z^4+_Z^2
+_Z+1,index=2)^2*RootOf(_Z^4+_Z^2+_Z+1,index=1)+359/257*_alpha^2*RootOf(_Z^4+_Z^2+_Z+1,index=2)*RootOf(_Z^4+_Z
^2+_Z+1,index=1)^3-74/257*_alpha^2*RootOf(_Z^4+_Z^2+_Z+1,index=2)*RootOf(_Z^4+_Z^2+_Z+1,index=1)^2+209/257*_al
pha^2*RootOf(_Z^4+_Z^2+_Z+1,index=1)*RootOf(_Z^...
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^4-x-2)/(x^4+x+1)/(x^4+x^2+x+1)^(1/2),x, algorithm="maxima")
[Out]
integrate((2*x^4 - x - 2)/(sqrt(x^4 + x^2 + x + 1)*(x^4 + x + 1)), x)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (16) = 32\).
time = 0.37, size = 35, normalized size = 1.94 \begin {gather*} \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} + x + 1} x + x + 1}{x^{4} + x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^4-x-2)/(x^4+x+1)/(x^4+x^2+x+1)^(1/2),x, algorithm="fricas")
[Out]
log((x^4 + 2*x^2 - 2*sqrt(x^4 + x^2 + x + 1)*x + x + 1)/(x^4 + x + 1))
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x**4-x-2)/(x**4+x+1)/(x**4+x**2+x+1)**(1/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^4-x-2)/(x^4+x+1)/(x^4+x^2+x+1)^(1/2),x, algorithm="giac")
[Out]
integrate((2*x^4 - x - 2)/(sqrt(x^4 + x^2 + x + 1)*(x^4 + x + 1)), x)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int -\frac {-2\,x^4+x+2}{\left (x^4+x+1\right )\,\sqrt {x^4+x^2+x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(x - 2*x^4 + 2)/((x + x^4 + 1)*(x + x^2 + x^4 + 1)^(1/2)),x)
[Out]
int(-(x - 2*x^4 + 2)/((x + x^4 + 1)*(x + x^2 + x^4 + 1)^(1/2)), x)
________________________________________________________________________________________